In the region \(\lbrace  r \ge R \rbrace \)  (where \(R\)  was chosen in the definition of the \(\Sigma \)  hypersurfaces) we have the following estimate (which is a combination of a Morawetz estimate and a redshift estimate, see [1]}): there exists a uniform constant \(C>0\) , such that for all \(u_0<\tau _1<\tau _2\)
\(\int _{\tau _1}^{\tau _2}\left(\int _{\Sigma _{\tau }\setminus \mathcal {N}_{\tau }}J^N[ \psi ]\cdot n_{\tau }\,d\mu _{\Sigma _{\tau }}\right)\,d\tau \le C \int _{\Sigma _{\tau _1}}( J^N[\psi ]\cdot n_{\tau _1}+J^N[T\psi ]\cdot n_{\tau _1} )\,d\mu _{\Sigma _{\tau _1}}.\) 
